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Quartonic family: Difference between revisions

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== Quartonic ==
== Quartonic ==
The name ''quartonic'' means quarter-tone, which is the [[generator]] of this temperament.
The name ''quartonic'' refers to the [[quartertone]], since a somewhat flattened quartertone (representing the triptolemaic chromatic semitone, i.e. [[porcupine comma]]) is the [[generator]] of this temperament. The [[ploidacot]] for the temperament is omega-hendecacot, with eleven generator steps giving the [[4/3|perfect fourth]]. It is a member of the [[schismic–Mercator equivalence continuum]], with equivalence number ''n'' = 11/2.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 10: Line 10:


{{Mapping|legend=1| 1 2 3 | 0 -11 -18 }}
{{Mapping|legend=1| 1 2 3 | 0 -11 -18 }}
: mapping generators: ~2, ~250/243


[[Optimal tuning]] ([[CTE]]): 2 = 1200.0000{{c}}, ~250/243 = 45.2368{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9709{{c}}, ~250/243 = 45.2318{{c}}
: [[error map]]: {{val| -0.029 +0.437 -0.574 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~250/243 = 45.2349{{c}}
: error map: {{val| 0.000 +0.461 -0.541 }}


{{Optimal ET sequence|legend=1| 26, 27, 53, 239, 292, 345, 398, 451 }}
{{Optimal ET sequence|legend=1| 26, 27, 53, 239, 292, 345, 398, 451 }}


[[Badness]] (Smith): 0.117250
[[Badness]] (Sintel): 2.75


=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the [[Normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
The second comma of the [[Normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
* [[1728/1715]] or [[4000/3969]] gives septimal quartonic, with interpretation of the generator ~36/35. It also tempers out [[4375/4374]].  
* [[1728/1715]] or [[4000/3969]] gives septimal quartonic, with interpretation of the generator ~36/35. It also tempers out [[4375/4374]].  
* [[Hemimage comma|10976/10935]] gives yarman I (80 & 159) and slices the quartonic generator in three.
* [[10976/10935]] gives yarman I (80 & 159) and slices the quartonic generator in three.
* 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.  
* 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.  
* [[2401/2400]] gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
* [[2401/2400]] gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
* [[Landscape comma|250047/250000]] gives triquart (27 & 159) with 1/3-octave period.
* [[250047/250000]] gives triquart (27 & 159) with 1/3-octave period.
* [[Dimcomp comma|390625/388962]] or [[canousma|4802000/4782969]] gives quartiquart (80 & 212) with 1/4-octave period.
* [[390625/388962]] or [[4802000/4782969]] gives quartiquart (80 & 212) with 1/4-octave period.
* [[Mirkwai comma|16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.
* [[16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.


== Septimal quartonic ==
== Septimal quartonic ==
Septimal quartonic tempers out the [[orwellisma]], the [[octagar comma]], as well as the [[ragisma]], and may be described as the {{nowrap| 26 & 27 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 34: Line 41:
{{Mapping|legend=1| 1 2 3 3 | 0 -11 -18 -5 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -11 -18 -5 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~36/35 = 45.2652{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1424{{c}}, ~36/35 = 45.1064{{c}}
: [[error map]]: {{val| -0.858 +0.160 -0.801 +3.069 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 45.1881{{c}}
: error map: {{val| 0.000 +0.976 +0.301 +5.234 }}


{{Optimal ET sequence|legend=1| 26, 27, 53, 80, 133d, 186d, 319dd }}
{{Optimal ET sequence|legend=1| 26, 27, 53, 80, 133d, 186d, 319dd }}


[[Badness]] (Smith): 0.042632
[[Badness]] (Sintel): 1.08


=== 11-limit ===
=== 11-limit ===
Line 47: Line 58:
Mapping: {{mapping| 1 2 3 3 5 | 0 -11 -18 -5 -41 }}
Mapping: {{mapping| 1 2 3 3 5 | 0 -11 -18 -5 -41 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.1674{{c}}
Optimal tunings:
* WE: ~2 = 1198.8787{{c}}, ~36/35 = 44.9994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.0863{{c}}


{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80 }}
{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80 }}


Badness (Smith): 0.034031
Badness (Sintel): 1.13


==== 13-limit ====
==== 13-limit ====
Line 60: Line 73:
Mapping: {{mapping| 1 2 3 3 5 4 | 0 -11 -18 -5 -41 -8 }}
Mapping: {{mapping| 1 2 3 3 5 4 | 0 -11 -18 -5 -41 -8 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.1632{{c}}
Optimal tunings:
* WE: ~2 = 1199.2991{{c}}, ~36/35 = 45.0539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.1049{{c}}


{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80, 133d, 186d }}
{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80, 133d }}


Badness (Smith): 0.023875
Badness (Sintel): 0.987


=== Quarto ===
=== Quarto ===
Line 73: Line 88:
Mapping: {{mapping| 1 2 3 3 4 | 0 -11 -18 -5 -14 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -11 -18 -5 -14 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.4022{{c}}
Optimal tunings:
* WE: ~2 = 1198.3094{{c}}, ~36/35 = 45.0688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.2329{{c}}


{{Optimal ET sequence|legend=0| 26, 53e, 132ee }}
{{Optimal ET sequence|legend=0| 26, 27e, 53e, 80ee }}


Badness (Smith): 0.041786
Badness (Sintel): 1.38


==== 13-limit ====
==== 13-limit ====
Line 86: Line 103:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -11 -18 -5 -14 -8 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -11 -18 -5 -14 -8 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.3857{{c}}
Optimal tunings:
* WE: ~2 = 1198.9315{{c}}, ~36/35 = 45.1650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.2640{{c}}


{{Optimal ET sequence|legend=0| 26, 53e, 132ee }}
{{Optimal ET sequence|legend=0| 26, 27e, 53e }}


Badness (Smith): 0.027692
Badness (Sintel): 1.14


=== Quartz ===
=== Quartz ===
Line 99: Line 118:
Mapping: {{mapping| 1 2 3 3 3 | 0 -11 -18 -5 12 }}
Mapping: {{mapping| 1 2 3 3 3 | 0 -11 -18 -5 12 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 45.3313{{c}}
Optimal tunings:
* WE: ~2 = 1200.5082{{c}}, ~33/32 = 45.4047{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 45.3739{{c}}


{{Optimal ET sequence|legend=0| 26, 53 }}
{{Optimal ET sequence|legend=0| 26, 53 }}


Badness (Smith): 0.053285
Badness (Sintel): 1.76


==== 13-limit ====
==== 13-limit ====
Line 112: Line 133:
Mapping: {{mapping| 1 2 3 3 3 4 | 0 -11 -18 -5 12 -8 }}
Mapping: {{mapping| 1 2 3 3 3 4 | 0 -11 -18 -5 12 -8 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 45.3168{{c}}
Optimal tunings:
* WE: ~2 = 1200.5899{{c}}, ~33/32 = 45.4096{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 45.3739{{c}}


{{Optimal ET sequence|legend=0| 26, 53 }}
{{Optimal ET sequence|legend=0| 26, 53 }}


Badness (Smith): 0.028818
Badness (Sintel): 1.19


=== Biquartonic ===
=== Biquartonic ===
Line 124: Line 147:


Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}
Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}
: mapping generators: ~99/70, ~36/35


Optimal tuning (CTE): ~99/70 = 600.0000{{c}}, ~36/35 = 45.2678{{c}}
Optimal tunings:
* WE: ~99/70 = 599.5389{{c}}, ~36/35 = 45.0947{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~36/35 = 45.1692{{c}}


{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54c, 80, 186de, 266dde }}


Badness (Smith): 0.060737
Badness (Sintel): 2.01


==== 13-limit ====
==== 13-limit ====
Line 138: Line 164:
Mapping: {{mapping| 2 4 6 6 7 8 | 0 -11 -18 -5 -1 -8 }}
Mapping: {{mapping| 2 4 6 6 7 8 | 0 -11 -18 -5 -1 -8 }}


Optimal tuning (CTE): ~55/39 = 600.0000{{c}}, ~40/39 = 45.2544{{c}}
Optimal tunings:
* WE: ~55/39 = 599.6870{{c}}, ~40/39 = 45.1293{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~40/39 = 45.1789{{c}}


{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}


Badness (Smith): 0.039891
Badness (Sintel): 1.65


==== 17-limit ====
==== 17-limit ====
Line 151: Line 179:
Mapping: {{mapping| 2 4 6 6 7 8 9 | 0 -11 -18 -5 -1 -8 -11 }}
Mapping: {{mapping| 2 4 6 6 7 8 9 | 0 -11 -18 -5 -1 -8 -11 }}


Optimal tuning (CTE): ~17/12 = 600.0000{{c}}, ~34/33 = 45.2397{{c}}
Optimal tunings:
* WE: ~17/12 = 599.7706{{c}}, ~40/39 = 45.1427{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~40/39 = 45.1790{{c}}


{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}


Badness (Smith): 0.028112
Badness (Sintel): 1.43


==== 19-limit ====
==== 19-limit ====
Line 164: Line 194:
Mapping: {{mapping| 2 4 6 6 7 8 9 10 | 0 -11 -18 -5 -1 -8 -11 -20 }}
Mapping: {{mapping| 2 4 6 6 7 8 9 10 | 0 -11 -18 -5 -1 -8 -11 -20 }}


Optimal tuning (CTE): ~17/12 = 600.000{{c}}, ~39/38 = 45.222{{c}}
Optimal tunings:
* WE: ~17/12 = 599.7848{{c}}, ~39/38 = 45.1288{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~39/38 = 45.1634{{c}}


{{Optimal ET sequence|legend=0| 26, 54ch, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54ch, 80, 106 }}


Badness (Smith): 0.0213
Badness (Sintel): 1.30


=== Yarm ===
=== Yarm ===
Line 177: Line 209:
Mapping: {{mapping| 1 2 3 3 4 | 0 -33 -54 -15 -43 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -33 -54 -15 -43 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0880{{c}}
Optimal tunings:
* WE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0363{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0617{{c}}


{{Optimal ET sequence|legend=0| 79, 80, 159d }}
{{Optimal ET sequence|legend=0| 79, 80, 239dd }}


Badness (Smith): 0.099950
Badness (Sintel): 3.30


==== 13-limit ====
==== 13-limit ====
Line 190: Line 224:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -33 -54 -15 -43 -24 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -33 -54 -15 -43 -24 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0842{{c}}
Optimal tunings:
* WE: ~2 = 1199.5798{{c}}, ~100/99 = 15.0551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0681{{c}}


{{Optimal ET sequence|legend=0| 79, 80, 159d }}
{{Optimal ET sequence|legend=0| 79, 80 }}


Badness (Smith): 0.061645
Badness (Sintel): 2.55


==== 17-limit ====
==== 17-limit ====
Line 203: Line 239:
Mapping: {{mapping| 1 2 3 3 4 4 4 | 0 -33 -54 -15 -43 -24 7 }}
Mapping: {{mapping| 1 2 3 3 4 4 4 | 0 -33 -54 -15 -43 -24 7 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0840{{c}}
Optimal tunings:
* WE: ~2 = 1199.6800{{c}}, ~100/99 = 15.0624{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0705{{c}}


{{Optimal ET sequence|legend=0| 79, 80, 159d }}
{{Optimal ET sequence|legend=0| 79, 80 }}


Badness (Smith): 0.046718
Badness (Sintel): 2.38


== Yarman I ==
== Yarman I ==
Yarman I is the first interpretation of [[Ozan Yarman]]'s proposed tuning for the Turkish makam.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 215: Line 255:


{{Mapping|legend=1| 1 2 3 4 | 0 -33 -54 -95 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -33 -54 -95 }}
: mapping generators: ~2, ~126/125


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~126/125 = 15.0714{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8726{{c}}, ~126/125 = 15.0651{{c}}
: [[error map]]: {{val| -0.127 +0.641 -0.213 -0.523 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~126/125 = 15.0689{{c}}
: error map: {{val| 0.000 +0.771 -0.036 -0.374 }}


{{Optimal ET sequence|legend=1| 79d, 80, 159, 239, 398, 637 }}
{{Optimal ET sequence|legend=1| 79d, 80, 159, 239, 637, 876b }}


[[Badness]] (Smith): 0.193315
[[Badness]] (Sintel): 4.89


=== 11-limit ===
=== 11-limit ===
Line 229: Line 274:
Mapping: {{mapping| 1 2 3 4 4 | 0 -33 -54 -95 -43 }}
Mapping: {{mapping| 1 2 3 4 4 | 0 -33 -54 -95 -43 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0724{{c}}
Optimal tunings:
* WE: ~2 = 1199.8339{{c}}, ~100/99 = 15.0637{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0685{{c}}


{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398, 637, 1035bd }}
{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 876be }}


Badness (Smith): 0.049170
Badness (Sintel): 1.63


=== 13-limit ===
=== 13-limit ===
Line 242: Line 289:
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -33 -54 -95 -43 -24 }}
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -33 -54 -95 -43 -24 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0707{{c}}
Optimal tunings:
* WE: ~2 = 1200.1080{{c}}, ~100/99 = 15.0766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0737{{c}}


{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f, 637ff }}
{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f }}


Badness (Smith): 0.040929
Badness (Sintel): 1.69


=== 17-limit ===
=== 17-limit ===
Line 255: Line 304:
Mapping: {{mapping| 1 2 3 4 4 4 4 | 0 -33 -54 -95 -43 -24 7 }}
Mapping: {{mapping| 1 2 3 4 4 4 4 | 0 -33 -54 -95 -43 -24 7 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0706{{c}}
Optimal tunings:
* WE: ~2 = 1200.0222{{c}}, ~100/99 = 15.0718{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0713{{c}}


{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f, 637ff }}
{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f }}


Badness (Smith): 0.031015
Badness (Sintel): 1.58


=== 19-limit ===
=== 19-limit ===
Line 268: Line 319:
Mapping: {{mapping| 1 2 3 4 4 4 4 5 | 0 -33 -54 -95 -43 -24 7 -60 }}
Mapping: {{mapping| 1 2 3 4 4 4 4 5 | 0 -33 -54 -95 -43 -24 7 -60 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0683{{c}}
Optimal tunings:
* WE: ~2 = 1200.0698{{c}}, ~100/99 = 15.0722{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0706{{c}}


{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 637ffh }}
{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 398fh }}


Badness (Smith): 0.023193
Badness (Sintel): 1.41


=== 23-limit ===
=== 23-limit ===
Line 281: Line 334:
Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 | 0 -33 -54 -95 -43 -24 7 -60 -38 }}
Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 | 0 -33 -54 -95 -43 -24 7 -60 -38 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0676{{c}}
Optimal tunings:
* WE: ~2 = 1200.0946{{c}}, ~100/99 = 15.0732{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0710{{c}}


{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 637ffhi }}
{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 398fh }}


Badness (Smith): 0.017682
Badness (Sintel): 1.27


=== 29-limit ===
=== 29-limit ===
Line 294: Line 349:
Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 6 | 0 -33 -54 -95 -43 -24 7 -60 -38 -91 }}
Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 6 | 0 -33 -54 -95 -43 -24 7 -60 -38 -91 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0667{{c}}
Optimal tunings:
* WE: ~2 = 1200.0995{{c}}, ~100/99 = 15.0726{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0703{{c}}


{{Optimal ET sequence|legend=0| 79dhj, 80, 159, 239 }}
{{Optimal ET sequence|legend=0| 79dhj, 80, 159, 239, 398fh }}


Badness (Smith): 0.014289
Badness (Sintel): 1.19


== Yarman II ==
== Yarman II ==
Yarman II is the other interpretation of [[Ozan Yarman]]'s proposed tuning for the Turkish makam.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 306: Line 365:


{{Mapping|legend=1| 1 2 3 2 | 0 -33 -54 64 }}
{{Mapping|legend=1| 1 2 3 2 | 0 -33 -54 64 }}
: mapping generators: ~2, ~875/864


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~875/864 = 15.0995{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4734{{c}}, ~875/864 = 15.1122{{c}}
: [[error map]]: {{val| +0.473 +0.291 -0.950 -0.701 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~875/864 = 15.1048{{c}}
: error map: {{val| 0.000 -0.412 -1.971 -2.121 }}


{{Optimal ET sequence|legend=1| 79, 159 }}
{{Optimal ET sequence|legend=1| 79, 159, 397cd, 556cd, 715ccdd }}


[[Badness]] (Smith): 0.655487
[[Badness]] (Sintel): 16.6


=== 11-limit ===
=== 11-limit ===
Line 320: Line 384:
Mapping: {{mapping| 1 2 3 2 4 | 0 -33 -54 64 -43 }}
Mapping: {{mapping| 1 2 3 2 4 | 0 -33 -54 64 -43 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0982{{c}}
Optimal tunings:
* WE: ~2 = 1200.4357{{c}}, ~100/99 = 15.1126{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.1054{{c}}


{{Optimal ET sequence|legend=0| 79, 159 }}
{{Optimal ET sequence|legend=0| 79, 159, 397cd, 556cd }}


Badness (Smith): 0.143477
Badness (Sintel): 4.74


=== 13-limit ===
=== 13-limit ===
Line 333: Line 399:
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -33 -54 64 -43 -24 }}
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -33 -54 64 -43 -24 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0952{{c}}
Optimal tunings:
* WE: ~2 = 1200.5028{{c}}, ~100/99 = 15.1135{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.1052{{c}}


{{Optimal ET sequence|legend=0| 79, 159 }}
{{Optimal ET sequence|legend=0| 79, 159, 397cdff, 556cdff }}


Badness (Smith): 0.068150
Badness (Sintel): 2.82


=== 17-limit ===
=== 17-limit ===
Line 346: Line 414:
Mapping: {{mapping| 1 2 3 2 4 4 4 | 0 -33 -54 64 -43 -24 7 }}
Mapping: {{mapping| 1 2 3 2 4 4 4 | 0 -33 -54 64 -43 -24 7 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0950{{c}}
Optimal tunings:
* WE: ~2 = 1200.3840{{c}}, ~100/99 = 15.1086{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.1025{{c}}


{{Optimal ET sequence|legend=0| 79, 159 }}
{{Optimal ET sequence|legend=0| 79, 159 }}


Badness (Smith): 0.051019
Badness (Sintel): 2.60


== Tertiseptisix ==
== Tertiseptisix ==
Line 357: Line 427:
[[Comma list]]: 2401/2400, 390625000/387420489
[[Comma list]]: 2401/2400, 390625000/387420489


{{Mapping|legend=1| 1 13 21 15 | 0 -44 -72 -47 }}
{{Mapping|legend=1| 1 -31 -51 -32 | 0 44 72 47 }}
: mapping generators: ~2, ~1458/875


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~875/729 = 311.308{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0156{{c}}, ~1458/875 = 888.7028{{c}}
: [[error map]]: {{val| +0.016 +0.485 -0.507 -0.293 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1458/875 = 888.6915{{c}}
: error map: {{val| 0.000 +0.472 -0.523 -0.324 }}


{{Optimal ET sequence|legend=1| 27, 131bccd, 158cd, 185c, 212, 239, 451 }}
{{Optimal ET sequence|legend=1| 27, …, 212, 239, 451 }}


[[Badness]] (Smith): 0.155952
[[Badness]] (Sintel): 3.95


== Triquart ==
== Triquart ==
Line 371: Line 446:


{{Mapping|legend=1| 3 6 9 10 | 0 -11 -18 -14 }}
{{Mapping|legend=1| 3 6 9 10 | 0 -11 -18 -14 }}
: mapping generators: ~63/50, ~250/243


[[Optimal tuning]] ([[CTE]]): ~63/50 = 400.0000{{c}}, ~250/243 = 45.2083{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0521{{c}}, ~250/243 = 45.2373{{c}}
: [[error map]]: {{val| +0.156 +0.747 -0.117 -1.628 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~250/243 = 45.2204{{c}}
: error map: {{val| 0.000 +0.621 -0.281 -1.911 }}


{{Optimal ET sequence|legend=1| 27, 105cd, 132d, 159, 186, 345d }}
{{Optimal ET sequence|legend=1| 27, 105cd, 132d, 159, 345d, 506d }}


[[Badness]] (Smith): 0.170062
[[Badness]] (Sintel): 4.30


== Quartiquart ==
== Quartiquart ==
Line 384: Line 464:


{{Mapping|legend=1| 4 8 12 15 | 0 -11 -18 -25 }}
{{Mapping|legend=1| 4 8 12 15 | 0 -11 -18 -25 }}
: mapping generators: ~25/21, ~250/243


[[Optimal tuning]] ([[CTE]]): ~25/21 = 300.0000{{c}}, ~250/243 = 45.2411{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 299.9955{{c}}, ~250/243 = 45.2382{{c}}
: [[error map]]: {{val| -0.018 +0.389 -0.655 +0.152 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~250/243 = 45.2400{{c}}
: error map: {{val| 0.000 +0.405 -0.634 +0.174 }}


{{Optimal ET sequence|legend=1| 80, 132d, 212, 292, 504 }}
{{Optimal ET sequence|legend=1| 80, 132d, 212, 292, 504 }}


[[Badness]] (Smith): 0.199116
[[Badness]] (Sintel): 5.04


=== 11-limit ===
=== 11-limit ===
Line 398: Line 483:
Mapping: {{mapping| 4 8 12 15 17 | 0 -11 -18 -25 -21 }}
Mapping: {{mapping| 4 8 12 15 17 | 0 -11 -18 -25 -21 }}


Optimal tuning (CTE): ~25/21 = 300.0000{{c}}, ~77/75 = 45.2303{{c}}
Optimal tunings:
* WE: ~25/21 = 300.0140{{c}}, ~77/75 = 45.2398{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~77/75 = 45.2342{{c}}


{{Optimal ET sequence|legend=0| 80, 132de, 212, 292, 504e }}
{{Optimal ET sequence|legend=0| 80, 132de, 212, 292, 504e }}


Badness (Smith): 0.062450
Badness (Sintel): 2.06


=== 13-limit ===
=== 13-limit ===
Line 411: Line 498:
Mapping: {{mapping| 4 8 12 15 17 16 | 0 -11 -18 -25 -21 -8 }}
Mapping: {{mapping| 4 8 12 15 17 16 | 0 -11 -18 -25 -21 -8 }}


Optimal tuning (CTE): ~25/21 = 300.0000{{c}}, ~40/39 = 45.2243{{c}}
Optimal tunings:
* WE: ~25/21 = 300.0805{{c}}, ~40/39 = 45.2812{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~40/39 = 45.2517{{c}}


{{Optimal ET sequence|legend=0| 80, 132de, 212 }}
{{Optimal ET sequence|legend=0| 80, 132de, 212 }}


Badness (Smith): 0.045028
Badness (Sintel): 1.86


=== 17-limit ===
=== 17-limit ===
Line 424: Line 513:
Mapping: {{mapping| 4 8 12 15 17 16 18 | 0 -11 -18 -25 -21 -8 -11 }}
Mapping: {{mapping| 4 8 12 15 17 16 18 | 0 -11 -18 -25 -21 -8 -11 }}


Optimal tuning (CTE): ~25/21 = 300.000{{c}}, ~40/39 = 45.218{{c}}
Optimal tunings:
* WE: ~25/21 = 300.1108{{c}}, ~40/39 = 45.2990{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~40/39 = 45.2604{{c}}


{{Optimal ET sequence|legend=0| 52cdeg, 80, 132deg, 212g }}
{{Optimal ET sequence|legend=0| 80, 132deg, 212g }}


Badness (Smith): 0.0312
Badness (Sintel): 1.59


=== 19-limit ===
=== 19-limit ===
Line 437: Line 528:
Mapping: {{mapping| 4 8 12 15 17 16 18 20 | 0 -11 -18 -25 -21 -8 -11 20 }}
Mapping: {{mapping| 4 8 12 15 17 16 18 20 | 0 -11 -18 -25 -21 -8 -11 20 }}


Optimal tuning (CTE): ~25/21 = 300.000{{c}}, ~39/38 = 45.210{{c}}
Optimal tunings:
* WE: ~25/21 = 300.1212{{c}}, ~39/38 = 45.3015{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~39/38 = 45.2590{{c}}


{{Optimal ET sequence|legend=0| 52cdegh, 80, 132degh, 212gh }}
{{Optimal ET sequence|legend=0| 80, 132degh, 212gh }}


Badness (Smith): 0.0224
Badness (Sintel): 1.36


== Quintiquart ==
== Quintiquart ==
Line 449: Line 542:


{{Mapping|legend=1| 5 10 15 18 | 0 -11 -18 -21 }}
{{Mapping|legend=1| 5 10 15 18 | 0 -11 -18 -21 }}
: mapping generators: ~35721/31250, ~250/243


[[Optimal tuning]] ([[CTE]]): ~35721/31250 = 240.0000{{c}}, ~250/243 = 45.2563{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~35721/31250 = 239.9950{{c}}, ~250/243 = 45.2520{{c}}
: [[error map]]: {{val| -0.025 +0.223 -0.925 +0.792 }}
* [[CWE]]: ~35721/31250 = 240.0000{{c}}, ~250/243 = 45.2546{{c}}
: error map: {{val| 0.000 +0.244 -0.897 +0.826 }}


{{Optimal ET sequence|legend=1| 80, 185c, 265, 610d, 875cd }}
{{Optimal ET sequence|legend=1| 80, 185c, 265, 610d }}


[[Badness]] (Smith): 0.357387
[[Badness]] (Sintel): 9.04


=== 11-limit ===
=== 11-limit ===
Line 463: Line 561:
Mapping: {{mapping| 5 10 15 18 19 | 0 -11 -18 -21 -9 }}
Mapping: {{mapping| 5 10 15 18 19 | 0 -11 -18 -21 -9 }}


Optimal tuning (CTE): ~8019/7000 = 240.0000{{c}}, ~250/243 = 45.2624{{c}}
Optimal tunings:
* WE: ~8019/7000 = 239.9613{{c}}, ~250/243 = 45.2270{{c}}
* CWE: ~8019/7000 = 240.0000{{c}}, ~250/243 = 45.2460{{c}}


{{Optimal ET sequence|legend=0| 80, 185c, 265, 610de, 875cde }}
{{Optimal ET sequence|legend=0| 80, 185c, 265, 345 }}


Badness (Smith): 0.103496
Badness (Sintel): 3.42


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Quartonic family| ]] <!-- main article -->
[[Category:Quartonic family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Quartonic_family"